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Distributions & Random Variables
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A random variable is a variable that can take different values, each associated with some probability, arising from a random process or phenomenon. We usually denote random variables with a capital letter, such as X.
It can be discrete (taken from a finite set of values) or continuous (taking any value within an interval).
The probability distribution of a random variable tells us how likely each outcome is.
A discrete random variable takes from a finite set of values:
where each possible value has an associated probability.
The sum of the probabilities for all possible values {x1,x2,…xn} of a discrete random variable X equals 1. In symbols,
where U denotes the sample space.
Probability distributions of discrete random variables can be given in a table or as an expression. As a table, distributions have the form
where the values in the row P(X=x) sum to 1.
Probability distributions can be given in a table or as an expression. As an expression, distributions have the form
for any discrete random variable X.
The expected value of a discrete random variable X is the average value you would get if you carried out infinitely many repetitions. It is a weighted sum of all the possible values:
The expected value is often denoted μ.
The variance of a discrete random variable can calculated using the formula
In probability, a game is a scenario where a player has a chance to win rewards based on the outcome of a probabilistic event. The rewards that a player can earn follow a probability distribution, for example X, that governs the likelihood of winning each reward. The expected return is the reward that a player can expect to earn, on average. It is given by E(X), where X is the probability distribution of the rewards.
Games can also have a cost, which is the price a player must pay each time before playing the game. If the cost is equal to the expected return, the game is said to be fair.
If you add a constant b to every possible value a random variable can take on, its expected value will increase by that constant, reflecting a "shift" in the random variable's distribution. Its variance will remain unchanged, since adding a constant has no impact on the "spread." Multiplying by a constant a scales both expectation and variance:
The binomial distribution models situations where the same action is repeated multiple times, each with the same chance of success. It has two key numbers: the number of attempts (n) and the probability of success in each attempt (p).
If a random variable X follows a binomial distribution, we write X∼B(n,p).
The binomial probability density function (aka pdf) is a function that models the likelihood of obtaining k successes from n trials where the likelihood of success of each trial is p. We calculate the probability of exactly k successes in n trials, P(X=k), using the calculator's binompdf function.
Press 2nd → distr → binompdf(. Once in the binompdf function, write your n value after "trials," your p value after "p," and your k value after "x value." Then hit enter twice and the calculator will return the probability you are interested in.
The distr button is located above vars . Once in the distr menu, you can also click alpha → A to navigate to the binompdf function.
The binomial cumulative density function tells us the probability of obtaining k or fewer successes in n trials, each with a likelihood of success of p. We calculate the probability of less than or equal to k successes in n trials, P(X≤k), using the calculator's binomcdf function.
Press 2nd → distr → binomcdf(. Once in the binomcdf function, write your n value after "trials," your p value after "p," and your k value after "x value." Then hit enter twice and the calculator will return the probability you are interested in.
The distr button is located above vars . Once in the distr menu, you can also click alpha → B to navigate to the binomcdf function.
If X∼B(n,p), then
and
The normal distribution, often called the bell curve, is a symmetric, bell-shaped probability distribution widely used to model natural variability and measurement errors. It appears frequently in natural settings because averaging many small, independent effects tends to produce results that cluster around a central value, naturally forming a bell-shaped distribution.
The normal distribution is characterized by its mean, μ, and standard deviation, σ, which completely and uniquely describe both the central value and how "spread out" the curve is. By convention, we describing the normal distribution by writing X∼N(μ,σ2). Notice that σ2 is the variance, not the standard deviation.
The probability that X is less than a given value a, written P(X<a), is equal to the area under the curve to the left of x=a:
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It follows that the total area under the curve is 1, which is required as the probabilities must sum to 1.
Because of the symmetry of the normal distribution, we know that
Further, for any real number a,
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It is also useful (but not often required) to know the empirical rule:
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To calculate P(a<X<b) for X∼N(μ,σ2) on your GDC, press 2nd →distr (on top of vars) to open the probability distribution menu. Select normalcdf( with your cursor. Type the value of a after "lower," the value of b after "upper," the value of μ after "μ," and the value of √σ2=σ after σ (since the calculator asks for standard deviation, not variance). Then click enter twice and the calculator will return the value of P(a<X<b).
If you want to find a one-sided probability like P(a<X), enter the value ±1×1099 as the upper or lower bound.
Under the hood, the calculator is finding the area under the normal curve between x=a and x=b:
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The calculator can also perform inverse normal calculations. That is, given the mean μ, the standard deviation σ, and the probability P(X<a)=k, the calculator can find the value a.
On your GDC, press 2nd →distr (on top of vars) to open the probability distribution menu. Select invNorm( with your cursor. Type the value of k after "area," the value of μ after "μ," and the value of √σ2=σ after σ (since the calculator asks for standard deviation, not variance). Then click enter twice and the calculator will return the value of a.
Note the calculator specifically returns the value of the "left end" of the tail. To find the value of some b when given P(b<X)=k, enter the value of 1−k (the complement) as the area.
For a random variable X with X∼N(μ,σ2) and a specific value x in the probability distribution, the z-value of x is defined as
The z-value measures how many standard deviations a value x lies above or below the mean. Put differently,
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A continuous random variable X is a random variable that can take any value in a given interval. Since there are infinitely many possible values within any interval (finite or infinite), the probability of any specific value is 0. Instead, you have to consider the probability that the value of X will fall within some specific range.
This probability is the area under a curve:
The function f(x) is called the probability density function. Its values are not probabilities - since P(X=x)=0 - but instead an abstract measure of how "densely packed" the probability is around each point.
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Since the probability of any sample space must be equal to 1,
If a probability density function is only defined for an interval a≤X≤b, then it is zero everywhere else and the following integrals are equivalent:
For a probability density function g(x) where ∫−∞∞g(x)=1, finding the value of k such that ∫−∞∞kg(x)=1 is called normalizing the probability function.
The expected value of a continuous random variable is
Intuitively, this can be thought of as the "balance point" of the distribution, the weighted sum of all values of x.
The expected value is also known as the mean.
The variance of a continuous random variable X is given by
The standard deviation can be found by taking the square root of this result.
Suppose that the random variable X is scaled and shifted, producing the random variable aX+b. The expected value and variance of the resulting variable are
The median of a continuous random variable X is the value m that splits the distribution into two equal areas:
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The mode of a continuous random variable X is that value x that maximizes f(x). On a graph, this corresponds to the peak of the probability density function.
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Suppose that the random variable X is scaled and shifted, producing the random variable aX+b. The expected value and variance of the resulting variable are